Higher November 2017 Paper 2 Q27
27 Solve \(\quad \dfrac{x}{4} - \dfrac{2x}{x + 2} = 1\)
Give your solutions to 2 decimal places.
You must show your working. [6 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(x(x + 2)\) or \(x^2 + 2x\) or \(2x \times 4\) or \(8x\) or \(4(x + 2)\) or \(4x + 8\) | M1 | |
| \(x(x + 2)\) or \(x^2 + 2x\) and \(2x \times 4\) or \(8x\) and \(4(x + 2)\) or \(4x + 8\) | M1dep | oe eg \(\dfrac{x(x + 2) - 2x \times 4}{4(x + 2)}\) |
| \(x(x + 2) - 2x \times 4 = 4(x + 2)\) | M1dep | oe equation with fractions eliminated dep on M2 |
| \(x^2 - 10x - 8\) (\(= 0\)) | A1 | oe 3-term quadratic equation with terms collected |
| \(\dfrac{--10 \pm \sqrt{(-10)^2 - 4 \times 1 \times -8}}{2 \times 1}\) or \(\dfrac{10 \pm \sqrt{100 + 32}}{2}\) or \(\dfrac{10 \pm \sqrt{132}}{2}\) or \(5 \pm \sqrt{5^2 + 8}\) or \(5 \pm \sqrt{33}\) or [10.744, 10.745] and [−0.745, −0.744] | M1 | oe Correct for their 3-term quadratic Allow correct factorisation of their 3-term quadratic |
| 10.74 and −0.74 with \(x^2 - 10x - 8\) (\(= 0\)) oe seen | A1 | Must both be to 2 decimal places |
| Alternative method 2 (from \(\dfrac{x}{4} = 1 + \dfrac{2x}{x + 2}\)) | ||
| \(x(x + 2)\) or \(x^2 + 2x\) or \((x + 2) + 2x\) or \(3x + 2\) or \(12x + 8\) | M1 | |
| \(\dfrac{x(x + 2)}{4}\) or \(\dfrac{x^2 + 2x}{4}\) and \(\dfrac{x + 2 + 2x}{x + 2}\) or \(\dfrac{3x + 2}{x + 2}\) | M1dep | |
| \(x(x + 2) = 4(x + 2 + 2x)\) or \(x(x + 2) = 4(3x + 2)\) | M1dep | oe equation with fractions eliminated dep on M2 |
| \(x^2 - 10x - 8\) (\(= 0\)) | A1 | oe 3-term quadratic equation with terms collected |
| \(\dfrac{--10 \pm \sqrt{(-10)^2 - 4 \times 1 \times -8}}{2 \times 1}\) or \(\dfrac{10 \pm \sqrt{100 + 32}}{2}\) or \(\dfrac{10 \pm \sqrt{132}}{2}\) or \(5 \pm \sqrt{5^2 + 8}\) or \(5 \pm \sqrt{33}\) or [10.744, 10.745] and [−0.745, −0.744] | M1 | oe Correct for their 3-term quadratic Allow correct factorisation of their 3-term quadratic |
| 10.74 and −0.74 with \(x^2 - 10x - 8\) (\(= 0\)) oe seen | A1 | Must both be to 2 decimal places |
| Alternative method 3 (from \(\dfrac{x}{4} - 1 = \dfrac{2x}{x + 2}\)) | ||
| \(\dfrac{x - 4}{4}\) | M1 | |
| \((x - 4)(x + 2)\) or \(x^2 - 4x + 2x - 8\) or \(x^2 - 2x - 8\) and \(2x \times 4\) or \(8x\) | M1dep | |
| \((x - 4)(x + 2) = 2x \times 4\) or \(x^2 - 4x + 2x - 8 = 8x\) | M1dep | oe equation with fractions eliminated dep on M2 |
| \(x^2 - 10x - 8\) (\(= 0\)) | A1 | oe 3-term quadratic equation with terms collected |
| \(\dfrac{--10 \pm \sqrt{(-10)^2 - 4 \times 1 \times -8}}{2 \times 1}\) or \(\dfrac{10 \pm \sqrt{100 + 32}}{2}\) or \(\dfrac{10 \pm \sqrt{132}}{2}\) or \(5 \pm \sqrt{5^2 + 8}\) or \(5 \pm \sqrt{33}\) or [10.744, 10.745] and [−0.745, −0.744] | M1 | oe Correct for their 3-term quadratic Allow correct factorisation of their 3-term quadratic |
| 10.74 and −0.74 with \(x^2 - 10x - 8\) (\(= 0\)) oe seen | A1 | Must both be to 2 decimal places |
Additional guidance
| 10.74 and −0.74 from T & I or with no working | 6 marks |
| 10.74 or −0.74 from T & I or with no working | Zero |
| In quadratic formula, do not allow \(-10^2\) for \((-10)^2\) unless recovered |